〈 Berger | Dillon 〉

The Lorentz group SO(3,1) is a symmetry group of relativistic mechanics. Quantum mechanically, the representations of this group that are interesting are those of the double cover, SL(2,C). The smallest irreducible representation of this group gives the “spin-1/2” representation.

The double cover isn’t too hard to understand: SO(3,1) isn’t simply connected (not every loop can be contracted to a point). The Universal Cover (in this case, double cover) is the “smallest” group “containing” SO(3,1) that is simply connected. This happens to be SL(2,C).

Since overall phase doesn’t matter in quantum mechanics, we can look for representations that carry an extra phase, and these are called “projective” representations. It turns out that for most symmetries, the projective representations of a group can be written as regular...

...representations of its universal cover. This is the origin of why we study SU(2) instead of SO(3) in 3-dimensional systems. The rotation group SO(3) isn’t simply connected, and SU(2) is it’s universal cover, which is what is relevant in quantum mechanics.